REAL k-FLATS TANGENT TO QUADRICS IN ℝ^{n}

FRANK SOTTILE THORSTEN THEOBALD

Abstract. Let d_{k,n} and #_{k,n} denote the dimension and the degree of the Grassmannian 𝔾_{k,n}, respectively. For each 1 ≤ k ≤ n − 2 there are 2^{d_{k,n}} · #_{k,n} (a priori complex) k-planes in ℙ^{n} tangent to d_{k,n} general quadratic hypersurfaces in ℙ^{n}. We show that this class of enumerative problem is fully real, i.e., for 1 ≤ k ≤ n − 2 there exists a configuration of d_{k,n} real quadrics in (affine) real space ℝ^{n} so that all the mutually tangent k-flats are real.